Part 1: Ordering Fractions with Like Denominators

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Katie and Raj both have the same size submarine sandwich. Each sandwich is cut into 8 equal slices. Katie is going to eat 2 pieces of her sandwich. Raj is going to eat 4 pieces of his sandwich.

Using the virtual fraction applet, the following image can be constructed. Notice the slider can help students count out the fractions along the rectangle. Students should be able to identify that Katie will eat 2/8 of her sandwich and Raj will eat 4/8 of his sandwich. As the students use the arrows to shade in the rectangle, they will see the name of the fractional part of the whole on the right. They also are forced at this point to see the rectangles in an order which eventually should transfer to using a number line model.

Raj eats more of his sandwich. Keep in mind that students need to discuss that the sandwiches are the same size, so the "unit" is the same for both Katie and Raj. They also need to discuss that the sandwiches were cut into equal pieces.

Many students will be set in their thinking that since 2 is less than 4, Katie eats less food. While this is true for like denominators, students need to add into their thoughts and discussions that they are comparing the same size pieces of the sandwich.

Given that the manipulatives are virtual, ask students to open up the Pattern Block Applet and the Integer Rods Applet to see if they could represent the sandwiches with those manipulatives. If using the physical manipulatives, one can see that this could easily get out of control and student might be confused with all of the different pieces. With the virtual manipulatives, students can switch quickly from applet to applet and explore why one manipulative perhaps is better than another for their investigation. There is number sense thinking going on when one make a choice about which pieces to use out of a manipulative for an investigation.

Have students write out sentences and then write out the math symbols. This helps build the foundation for their problem solving skills later in Algebra.

One observation that can be made is that the part of the sandwich that Raj will eat is twice as big as Katie's portion. This can be seen with the virtual fractions and students should be drawn into writing out the math! They can start to establish that 2* 2/8 = 4/8. Students need to put this in perspective. Some students will stay with the familiar and write 2*2 = 4. Some will get frustrated because they are giving a correct logical answer with respect to whole number thinking. They need to be aware that we can say things from different perspectives and they need to be able to communicate from different points of view or different number sets. This needs to be established on a simple level. Students encounter different number types in life such as fractions, decimals, and percentages and if they do not have a good basic foundation about these changes of perspectives, they soon get lost and give up.

Part 1 Problems

Answers will vary. Look for an understanding of the unit in the problem. Make sure that the students clearly state the unit which is broken up into 5 pieces. Encourage students to rewrite their work to refine their thinking after discussion about their response. Students could talk about their stories in a group and each could explain the "math" in their story. This activity amounts to making up a "word problem" or "story problem" about the numbers. Invite students to think about where they might see these number comparison used outside of school.

Part 2: Ordering Fractions with Unlike Denominators

Judy and Jay both have the same size submarine sandwich. Judy's sandwich is cut into 12 equal pieces. Jay's sandwich is cut into 6 equal pieces. Judy will eat 5 pieces of her sandwich for lunch. Jay will eat 4 pieces of her sandwich for lunch.

This problem was chosen so that the denominators are factors or multiples of each other. This example will open the door to other discussions about fractions such as equivalent fractions.

1. Look at the "sandwiches" you made.
   • Which fraction is less than one half?
   • Which fraction is greater than one half?

   Add other rectangles to explore this question. Use the slider to compare the fractions. Looking at the fraction as numbers, write an explanation of why one of the fractions is less than one half and one is greater than one half.
   

   As seen in the picture above, students can compare the position of 5/12 and 4/6 with respect to one half. Students can add another rectangle and use the slider to see that the quantities are to either side of one half. Recall that the NCTM standards call for students to learn to compare fractions by using a simple known fractional quantity such as one half at this level.

   Open a discussion about the numerators and denominators of these fractions. Ask how many 12ths are needed to build up to 1/2 on the their rectangle. Engage them in a discussion which brings out the fact that 6 of the 12ths are need to make 1/2 so then they can reason 5 of the 12ths certainly is smaller than 1/2 or symbolically 5/12 < 1/2. Similarly, repeat the discussion for 4/6.

   Notice, at this point, the rectangles are helping students move to see these fractions as on a number line.

   






2. If Judy and Jay combined their pieces of their sandwiches, would one sandwich have been enough for their lunch? Use the rectangles and write an explanation of your answer. Will they have any part of the sandwich left over?

   Again, the fractional pieces do not move but students can see that 2 of the 12ths line up with 1 of the 6ths using the slider. If they count off along one of the "sandwiches," they will see that 5/12 + 4/6 seems to be 1 and 1/12 or 13/12. One sandwich will not be enough! The structure of this problem was carefully chosen. This gives students a picture of why "units" will have to be broken into same sizes in order to add (or subtract). Notice they can do these types of problems using manipulatives to reason out the answer without the algorithm. If we gave a problem that didn't "line up," students would not be able to use simple reasoning from their whole number sense. Here they just used that 6 * 2 = 12. Doing several problems of this structure can give students the opportunity to come up with a reasonable way to add fractions. Moving later to problems where the denominators "don't match up" in terms of the rectangles, opens the door later to discuss how the factors of the denominators will become important in the addition or subtraction of fractions.

   Again, an interesting activity would be to have students model this problem with the Pattern Blocks Applet and the Integer Rods Applet to see if they could build these sandwiches!

   Notice, it is not an easy task to have physical strips line up and stay in position for a discussion. This is another benefit of the virtual manipulatives. This applet is designed so that the fractional parts of the rectangle do not move. We can put in place an option where the fractional parts will move. Some of the design features being OFF or ON help to focus the investigations which can be good for the classroom. We can design the virtual manipulative to meet instructional needs and to focus activities whereas the physical models cannot be controlled. These are very interesting differences in the manipulatives and these differences lead to new teaching techniques that will, hopefully, reach more learning styles.

Part 2 Problems

• Use the virtual fractions to compare the following fractions. Draw the image below each problem.
• Write >, <, or = in the box.
• Write a sentence using the words greater than, less than or equal to make your comparison.

Answers will vary. Look for an understanding of the unit in the problem. Make sure that the students clearly state which unit is broken up into 10 pieces and which unit is broken into 6 pieces. Encourage students to rewrite their work to refine their thinking after discussion about their response. Students could talk about their stories in a group and each could explain the "math" in their story. This activity amounts to making up a "word problem" or "story problem" about the numbers. Invite students to think about where they might see these number comparison used outside of school. Note: The denominators in this problem do not "match up." Invite students to include addition into their story and see how they approach this problem.

These fractions are equivalent. Give students other equivalent fractions to see using the applet. Ask them to reason out when two fractions will be equivalent. Depending on grade level, this can include how "write the math!"